If you consider the damping force is friction like in:
then the force should be $$F=\mu N$$ where $\mu$ is the coefficient of kinetic friction. Why then is the damping force assumed to be linearly dependent on velocity?
This question is actually one of the lab exercises I teach. For a spring-mass system, if the damping force is friction, then it is independent of velocity (verified experimentally). However, as mentioned in the comments, the damping force may not always be friction. For example, if the mass is a material like aluminium and it is oscillating over some magnets, the damping force will be linearly dependent on velocity.
Actually, this is just a simple model for resistive forces (usually duo to viscosity) , and in many situations you can not assume this force to be linearly dependent on velocity.(for example ,usually this model is correct only for small enough objects)
In many everyday examples, this force is due to viscous forces. If you consider an (small enough) object moving in a viscose fluid ,and if the speed of the object is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from $v$ in the vicinity of the object to zero in farther points. Each layer of fluid will move faster than the one just outer , and friction between them will give rise to a force resisting the motion of object. (a force in the direction opposite to its motion.)
There is a formula (named Stokes' law) for the frictional force on such a small (spherical) particle moving with constant(terminal) speed in a viscose fluid :
$F = 6 \pi\mu Rv$
where $v$ is the particle's velocity and $R$ is the radius of the spherical object and $\mu$ is a measure of fluid's viscosity.(named dynamic viscosity)
A linear model for resistive forces may be used in other contexts too; One can model laser cooling process as a linear resistive force against the motion of microscopic particles.
For more quantitative formulas about viscous forces you can see Viscosity in Wikipedia.